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A $C^0$-theory for the blow-up of second order elliptic equations of critical Sobolev growth
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## A $C^0$-theory for the blow-up of second order elliptic equations of critical Sobolev growth

### Olivier Druet, Emmanuel Hebey, and Frédéric Robert

**Abstract.**
Let $(M,g)$ be a smooth compact Riemannian manifold
of dimension $n \ge 3$, and $\Delta_g = -div_g\nabla$ the Laplace-Beltrami
operator. Also let $2^\star$ be the critical Sobolev exponent
for the embedding of the Sobolev space $H_1^2(M)$ into Lebesgue spaces,
and $h$ a smooth function on $M$.
Elliptic equations of critical Sobolev growth like
\[\Delta_gu + hu = u^{2^\star-1}\]
have been the target of investigation
for decades. A very nice $H_1^2$-theory for the
asymptotic behaviour of solutions of such an equation
is available since the 1980's.
In this announcement we present the $C^0$-theory we have recently developed.
Such a theory provides
sharp pointwise estimates for the asymptotic behaviour of solutions of the
above equation.

*Copyright 2003 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**09** (2003), pp. 19-25
- Publisher Identifier: S 1079-6762(03)00108-2
- 2000
*Mathematics Subject Classification*. Primary 35J60; Secondary 58J05
*Key words and phrases*. Critical elliptic equations, blow-up behaviour, bubbles
- Received by editors November 4, 2002
- Received by editors in revised form December 16, 2002
- Posted on February 3, 2003
- Communicated by Tobias Colding
- Comments (When Available)

**Olivier Druet**

Département de Mathématiques, Ecole Normale Supérieure de Lyon, 46 allée d'Italie, 69364 Lyon cedex 07, France

*E-mail address:* `Olivier.Druet@umpa.ens-lyon.fr`

**Emmanuel Hebey**

Département de Mathématiques, Université de Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France

*E-mail address:* `Emmanuel.Hebey@math.u-cergy.fr`

**Frédéric Robert**

Department of Mathematics, ETH Zürich, CH-8092 Zürich, Switzerland

*E-mail address:* `Frederic.Robert@math.ethz.ch`

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